The electronic contribution to the polarization can be obtained from
\(Z_q\)  by taking the first finite difference derivative with respect to
\(q\) . Indeed, the electronic contribution to the many-body
polarization expression derived by Resta [1]} for
interacting systems can be derived by taking the first finite
difference derivative of \(\ln Z_q\)  at \(q=0\) ,
\(\langle X \rangle = i \frac{L}{2 \pi } \left(\ln Z_1 - \ln Z_0\right) = \frac{L}{2 \pi } \mbox{Im} \ln Z_1.\) 
